Is it possible to parent an object to a different bone in 2 different (multiple) animation?
The only solution is the Child_Of constraint if you want to alter parenting in pose mode. If that leads to another problem, ask it, there is probably solution to it.

Make sure the bone that is gonna get the Child_Of constraint has no parents. Else you will get multiple parent transformations (the original parent and the constraint on top).

There is a way to rig an inverse parent transformation to negate the original parent (if present) when needed from pose mode, but it is complicated and probably not worth it.